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Chapter 13: Problem 2

The flow rate in a device used for air quality measurement depends on the pressure drop \(x\) (inches of water) across the device's filter. Suppose that for \(x\) values between 5 and 20 , these two variables are related according to the simple linear regression model with true regression line \(y=-0.12+0.095 x\). a. What is the true average flow rate for a pressure drop of 10 in.? A drop of 15 in.? b. What is the true average change in flow rate associated with a 1 -in. increase in pressure drop? Explain. c. What is the average change in flow rate when pressure drop decreases by 5 in.?

Short Answer

Expert verified
a. The true average flow rates for pressure drops of 10 and 15 in. are 0.83 and 1.295 respectively. b. The true average change in flow rate associated with a 1-inch increase in pressure drop is 0.095. c. The change in flow rate when pressure drop decreases by 5 in. is -0.475.

Step by step solution

01

Calculate the True Average Flow Rate for a Pressure Drop of 10 in.

Plug \(x=10\) into the linear regression equation \(y=-0.12+0.095x\). The true average flow rate will be equal to the obtained \(y\) value: \(y=-0.12+0.095 \cdot 10\). Calculate the result.
02

Calculate the True Average Flow Rate for a Pressure Drop of 15 in.

Now, substitute \(x=15\) into the equation \(y=-0.12+0.095x\). The true average flow rate will be equal to this calculated \(y\) value: \(y=-0.12+0.095\cdot 15\). Solve for this value.
03

Calculate the True Average Change in Flow Rate Associated with a 1-in. Increase in Pressure Drop

The slope of the line in the simple linear regression equation indicates the change in \(y\) (flow rate) with each one unit increase in \(x\) (pressure drop). In this case, the slope is 0.095, thus the true average change in flow rate associated with a 1-inch increase in pressure drop is 0.095.
04

Calculate the Average Change in Flow Rate when Pressure Drop Decreases by 5 in.

Since it's given that the flow rate changes by 0.095 for every 1 inch of pressure drop, a decrease of 5 inches in pressure drop will change the flow rate by \(0.095 \cdot 5\). Calculate this product.

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